In 2730, y c1e3x c2ex 2x is a twoparameter family of the second-order DE y 2y 3y 6x 4

In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dy
dx  f(x, y). The symbol c1 represents a constant.

In Problems 1 and 2 fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol c1 and has the form dy
dx  f(x, y). The symbol c1 represents a constant.

In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y )  0. The symbols c1, c2, and k represent constants.

In Problems 3 and 4 fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y )  0. The symbols c1, c2, and k represent constants.

In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y )  0. The symbols c1 and c2 represent constants.

In Problems 5 and 6 compute y and y and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c1 and c2 and has the form F(y, y y )  0. The symbols c1 and c2 represent constants.

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 712 match each of the given differential equations with one or more of these solutions: (a) y  0, (b) y  2, (c) y  2x, (d) y  2x2 .

In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.

In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.

In Problems 15 and 16 interpret each statement as a differential equation.

In Problems 15 and 16 interpret each statement as a differential equation.

(a) Give the domain of the function y  x2/3. (b) Give the largest interval I of definition over which y  x2/3 is solution of the differential equation 3xy
2y  0.

(a) Verify that the one-parameter family y2
2y  x2
x c is an implicit solution of the differential equation (2y
2)y  2x
1. (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition y(0)  1. (c) Use your result in part (b) to find an explicit function y 
(x) that satisfies y(0)  1. Give the domain of the function
. Is y 
(x) a solution of the initial-value problem? If so, give its interval I of definition; if not, explain.

Given that y  x
2
x is a solution of the DE xy y  2x. Find x0 and the largest interval I for which y(x) is a solution of the first-order IVP xy y  2x, y(x0)  1.

Suppose that y(x) denotes a solution of the first-order IVP y  x2 y2 , y(1) 
1 and that y(x) possesses at least a second derivative at x  1. In some neighborhood of x  1 use the DE to determine whether y(x) is increasing or decreasing and whether the graph y(x) is concave up or concave down.

A differential equation may possess more than one family of solutions. (a) Plot different members of the families y 
1(x)  x2 c1 and y 
2(x) 
x2 c2. (b) Verify that y 
1(x) and y 
2(x) are two solutions of the nonlinear first-order differential equation (y)2  4x2 . (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).

What is the slope of the tangent line to the graph of a solution of that passes through (
1, 4)? In Pr

In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.

In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.

In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.

In Problems 2326 verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.

In Problems 2730, y  c1e3x c2e
x
2x is a twoparameter family of the second-order DE y
2y
3y  6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2

In Problems 2730, y  c1e3x c2e
x
2x is a twoparameter family of the second-order DE y
2y
3y  6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2

In Problems 2730, y  c1e3x c2e
x
2x is a twoparameter family of the second-order DE y
2y
3y  6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2

In Problems 2730, y  c1e3x c2e
x
2x is a twoparameter family of the second-order DE y
2y
3y  6x 4. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 2

The graph of a solution of a second-order initial-value problem d2y
dx2  f(x, y, y), y(2)  y0, y(2)  y1, is given in Figure 1.R.1. Use the graph to estimate the values of y0 and y1.

A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water and water leaks from a circular hole of radius inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole.

The number of field mice in a certain pasture is given by the function 200
10t, where time t is measured in years. Determine a differential equation governing a population of owls that feed on the mice if the rate at which the owl population grows is proportional to the difference between the number of owls at time t and number of field mice at time t.

Suppose that dA
dt 
0.0004332 A(t) represents a mathematical model for the radioactive decay of radium-226, where A(t) is the amount of radium (measured in grams) remaining at time t (measured in years). How much of the radium sample remains at the time t when the sample is decaying at a rate of 0.002 gram per year?